The Basis of a Knowledge Space and a Generalized Interval Order

Can one read off from the basis of a knowledge space in a simple way whether or not the space is ordinal? A positive answer can be derived from Birkhoffs Theorem: One regards the basis as an abstract partial order which is represented as a suborder of the Boolean lattice of subsets of the item set. If this set representation consists of the lower cones of the partial order then the knowledge space is ordinal (or quasi ordinal if slight modifications are observed). Some partial orders have the property that all their set representations which are parsimonious in a certain sense are equal to the representation by its lower cones. These orders allow only ordinal knowledge spaces. They are characterized by a special condition which generalizes the interval order condition.